Suppose the probability of a false positive result on a mammogram is 7% and that radiologists' interpretations of mammograms are mutually independent in the sense that whether or not a radiologist finds a positive result on one mammogram does not influence whether or not a radiologist finds a positive result on another mammogram. Assume that a woman has one mammogram every year for ten years. (a) What is the probability (as a %) that she will have no false positive results during that time?
(b) What is the probability (as a %) that she will have at least one false positive result during that time?
(c) What is the probability (as a %) that she will have exactly two false positive results during that time?
(d) Suppose that the probability of a false negative result on a mammogram is 3%, and assume that the probability that a randomly chosen woman has breast cancer is 0.0003.
(i) If a woman has a positive test result one year, what is the probability (as a %) that she actually has breast cancer?
(ii) If a woman has a negative test result one year, what is the probability (as a %) that she actually has breast cancer?

a) 48.4%

b) 51.6%

c) 12.3%

d) i) 93.0%  ii) 3.0%

Step-by-step explanation:

a)

P(fp) = probability of false positive result

P(fp) = 0.07

P(fp = 0) = (1 - 0.07)¹⁰

P(fp = 0) = 0.4839... ⇒ 48.4%

b)

P(fp = 1) = 1 - 0.4839...

P(fp = 1) = 0.516... ⇒ 51.6%

c)

P(fp = 2) = 10C2(0.07)²(0.93)⁸

P(fp = 2) = 0.123... ⇒ 12.3%

d)

P(fn) = probability of false negative

P(b) = probability a randomly selected woman has breast cancer  

P(fn) = 0.03

P(b) = 0.0003

If she has a positive test, it is either a true positive, meaning she actually has breast cancer, or a false positive, meaning she doesn't;

If P(fp) = 0.07, then the probability of a true positive is 0.93 or 93%

ii)

If she has a negative test, a true negative means she doesn't have breast cancer, and a false negative means she does have breast cancer;

The probability of a false negative is 0.03 or 3%

(I'm not entirely sure about part d, but I've answered as I've understood it according to my logic, they seem to be somewhat trick questions unless I've incorrectly understood the questions)

answer:   2 multiplied by x =24

step-by-step explanation:

answer: integers are just whole numbers: positive, negative, and zero.

i = {-∞, -2, -1, 0, 1, 2, 3, ∞}

let x = 1st integer

let y = 2nd integer

x = 3y + 17   {equation 1}

xy = -24         {equation 2}

substitute 3y+17 in place of x n equation 2

and solve for y.

(3y+17)y = -24

3y2 + 17y = -24

3y2 + 17y + 24 = 0

from quadratic equation:

y = [-17±√(172-4(3)(24))]/(2(3))

y = (-17 ±√1)/6

y = (-17+1)/6   or     y = (-17-1)/6

y = -16/6           or     y = -18/6

y = -2 2/3         or   y = -3

since we are dealing with

y = -3

x = 3y+17 = -9+17 = 8

the integers are -3, 8

step-by-step explanation: